Closed open set

In the sub-area of topology of mathematics , a closed open set (in English clopen set , in German also closed set ) is a subset of a topological space that is closed and open at the same time .

This seems strange at first glance; However, it should be remembered that the terms open and closed have a different meaning in topology than in everyday language. A set is closed when its complement is open, which gives the possibility of an open set whose complement is also open, whereby both sets are both open and closed and therefore closed and open. Similarly, a set is open when its complement is complete. It follows that a closed open set results when a set is closed and its complement is closed. The concept of the closed open set is not to be confused with that of the half-open interval .

Examples

In every topological space, the empty set and the whole space are closed and open. In a contiguous topological space , these are the only subsets that are closed and open.

In the topological space X , which consists of the union of the two closed intervals and , provided with the induced topology from the standard topology, the set is closed and open. ${\ displaystyle [0,1]}$ ${\ displaystyle [2,3]}$ ${\ displaystyle \ mathbb {R}}$ ${\ displaystyle [0,1]}$

Similarly, in the topological space Y , which consists of the union of the two open intervals and , provided with the induced topology from the standard topology, the set is closed and open. ${\ displaystyle (0,1)}$ ${\ displaystyle (2,3)}$ ${\ displaystyle \ mathbb {R}}$ ${\ displaystyle (0,1)}$

In general, a connected component of a space is not open and closed: In the Alexandroff compactification of the set of natural numbers , the infinitely distant point forms a connected component that is not open. ${\ displaystyle \ mathbb {N}}$

Consider the set of rational numbers with the standard topology, and therein the subset A of all rational numbers that are greater than (or here equivalently: at least as large as) the square root of 2. Since is irrational, one can easily show that A is closed and open. Note, however, that A as a subset of the real numbers is neither closed nor open; the set of all real numbers greater than is open but not closed, while the set of all real numbers at least as large as is closed but not open. ${\ displaystyle \ mathbb {Q}}$ ${\ displaystyle {\ sqrt {2}}}$ ${\ displaystyle {\ sqrt {2}}}$ ${\ displaystyle {\ sqrt {2}}}$

properties

A subset of a topological space is open and closed if and only if its edge is empty.
A topological space X is if and contiguous when the single sealed open sets the empty set and X are.
Every closed open subset can be represented as a (possibly infinite) union of connected components.
If every connected component is open (which is the case, for example, if X has only finitely many components, or if X is locally connected ), then every union of connected components is also closed and open.
A topological space is discrete if and only if every subset is closed and open.
For every topological space the closed open sets form a Boolean algebra .
An open subgroup of a topological group is also closed. A closed subset of finite index is also open.

literature

Viktor I. Ponomarev: Open-closed set . In: Michiel Hazewinkel (Ed.): Encyclopaedia of Mathematics . Springer-Verlag , Berlin 2002, ISBN 978-1-55608-010-4 (English, online ).

Individual evidence

↑ Gerd Laures, Markus Szymik: Connection and separation . Springer Spectrum, Berlin, Heidelberg 2015, ISBN 978-3-662-45952-2 , doi : 10.1007 / 978-3-662-45953-9_3 .

[1] Gerd Laures, Markus Szymik: Connection and separation . Springer Spectrum, Berlin, Heidelberg 2015, ISBN 978-3-662-45952-2 , doi : 10.1007 / 978-3-662-45953-9_3 .